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Simplifying 27x = 15x2 + -22x + 8 Reorder the terms: 27x = 8 + -22x + 15x2 Solving 27x = 8 + -22x + 15x2 Solving for variable 'x'. Reorder the terms: -8 + 27x + 22x + -15x2 = 8 + -22x + 15x2 + -8 + 22x + -15x2 Combine like terms: 27x + 22x = 49x -8 + 49x + -15x2 = 8 + -22x + 15x2 + -8 + 22x + -15x2 Reorder the terms: -8 + 49x + -15x2 = 8 + -8 + -22x + 22x + 15x2 + -15x2 Combine like terms: 8 + -8 = 0 -8 + 49x + -15x2 = 0 + -22x + 22x + 15x2 + -15x2 -8 + 49x + -15x2 = -22x + 22x + 15x2 + -15x2 Combine like terms: -22x + 22x = 0 -8 + 49x + -15x2 = 0 + 15x2 + -15x2 -8 + 49x + -15x2 = 15x2 + -15x2 Combine like terms: 15x2 + -15x2 = 0 -8 + 49x + -15x2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. 0.5333333333 + -3.266666667x + x2 = 0 Move the constant term to the right: Add '-0.5333333333' to each side of the equation. 0.5333333333 + -3.266666667x + -0.5333333333 + x2 = 0 + -0.5333333333 Reorder the terms: 0.5333333333 + -0.5333333333 + -3.266666667x + x2 = 0 + -0.5333333333 Combine like terms: 0.5333333333 + -0.5333333333 = 0.0000000000 0.0000000000 + -3.266666667x + x2 = 0 + -0.5333333333 -3.266666667x + x2 = 0 + -0.5333333333 Combine like terms: 0 + -0.5333333333 = -0.5333333333 -3.266666667x + x2 = -0.5333333333 The x term is -3.266666667x. Take half its coefficient (-1.633333334). Square it (2.667777780) and add it to both sides. Add '2.667777780' to each side of the equation. -3.266666667x + 2.667777780 + x2 = -0.5333333333 + 2.667777780 Reorder the terms: 2.667777780 + -3.266666667x + x2 = -0.5333333333 + 2.667777780 Combine like terms: -0.5333333333 + 2.667777780 = 2.1344444467 2.667777780 + -3.266666667x + x2 = 2.1344444467 Factor a perfect square on the left side: (x + -1.633333334)(x + -1.633333334) = 2.1344444467 Calculate the square root of the right side: 1.460973801 Break this problem into two subproblems by setting (x + -1.633333334) equal to 1.460973801 and -1.460973801.Subproblem 1
x + -1.633333334 = 1.460973801 Simplifying x + -1.633333334 = 1.460973801 Reorder the terms: -1.633333334 + x = 1.460973801 Solving -1.633333334 + x = 1.460973801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.633333334' to each side of the equation. -1.633333334 + 1.633333334 + x = 1.460973801 + 1.633333334 Combine like terms: -1.633333334 + 1.633333334 = 0.000000000 0.000000000 + x = 1.460973801 + 1.633333334 x = 1.460973801 + 1.633333334 Combine like terms: 1.460973801 + 1.633333334 = 3.094307135 x = 3.094307135 Simplifying x = 3.094307135Subproblem 2
x + -1.633333334 = -1.460973801 Simplifying x + -1.633333334 = -1.460973801 Reorder the terms: -1.633333334 + x = -1.460973801 Solving -1.633333334 + x = -1.460973801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.633333334' to each side of the equation. -1.633333334 + 1.633333334 + x = -1.460973801 + 1.633333334 Combine like terms: -1.633333334 + 1.633333334 = 0.000000000 0.000000000 + x = -1.460973801 + 1.633333334 x = -1.460973801 + 1.633333334 Combine like terms: -1.460973801 + 1.633333334 = 0.172359533 x = 0.172359533 Simplifying x = 0.172359533Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.094307135, 0.172359533}
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